ag.algebraic geometry - Question on an exercise in Hartshorne: Equivalence of categories

This is a slight reformulation of exercise II.5.9.(c) in Hartshorne's "Algebraic Geometry" which I don't understand.

Let $K$ be a field and $S=K[X_0,ldots,X_n]$ a graded ring. Set $X=Proj(S)$ and let $M$ be a graded $S$-module. The functors $Gamma_*$ definied by
$$
Gamma_*(mathcal{F})=bigoplus_{ninmathbb{Z}} (mathcal{F}(n))(X)
$$
and $~widetilde{phantom{cdot}}~$ (the "graded associated sheaf functor", see Hartshorne II.5. page 116 for a definition)
induce an equivalence of categories between the category $mathcal{A}$ of quasi-finitely generated (i.e. in relation to a finitely generated module) graded $S$-modules modulo a certain equivalence relation $approx$ and the category $mathcal{B}$ of coherent $mathcal{O}_X$-modules. The equivalence relation is: $Mapprox N$ if there is an integer $d$ such that $oplus_{kgeq d}M_kcongoplus_{kgeq d}N_k$.

I don't know what an "equivalence of categories" is in this context. Formally an "equivalence of categories" means in particular that there are isomorphisms $$hom_mathcal{A}(M,N)cong hom_mathcal{B}(widetilde{M},widetilde{N})$$ and $$hom_mathcal{B}(Y,Z)cong hom_mathcal{A}(Gamma_*(Y),Gamma_*(Z))$$
of sets. This is my problem: How is the sheaf $mathcal{H}om_mathcal{B}(Y,Z)$ considered as a set? Perhaps it should be $Gamma_*(mathcal{H}om_mathcal{B}(Y,Z))cong hom_mathcal{A}(Gamma_*(Y),Gamma_*(Z))$?